Plasmon-Polariton

56 4 CLEAN QUANTUM WIRE electric eld were not coupled, the transverse eld would not need to be neglected as it would simply not enter the equation determining the local longitudinal eld.

Hence, the anisotropy is neglected together with the local transverse eld. It is reasonable to neglect it then also in the evaluation of the local longitudinal eld by projecting the electric elds.

4.4 Local Field Eects 57 The elements of the vectors ^E~_{L and} ^E~_T are the local elds ~EL and ~ET evaluated at the discrete values qy = 4N;n=d and qz = 4M;m=d, see appendix A.2.1, where d is the diameter of the wire. The elements of the (3NM3NM) matrices^MAB(qx;!), where A and B denote L and T, are explicitly given in appendix A.2.1. The matrix

M

AB for A⁶=B is non-zero, hence longitudinal and transverse elds are coupled in the quantum wire in contrast to the example discussed in section 3.3.

When the external eld is either longitudinal or transverse, the discretized local eld equations become

M

AB

;1(qx;!)^E~_A(qx;!) =^E~_B;ext(qx;!): (162) How the i-th component of the discretized local eld vector^E~_Acan be obtained from eq. (162) is shown in appendix A.2.2. For completeness, the discretized dielectric function following from eq. (162) is given in appendix A.2.3.

Local-Field Eigenmodes

From eq. (162) and the theory on coupled linear equations, see e.g. [285], it is clear that a local eld may exist even in the absence of an external eld when the matrix

M

AB

;1is singular, i.e. its determinant is zero because then a system of homogeneous linear equations is solvable. Hence, in order to nd a local-eld eigenmode one has to search for the values of qx and ! for which

det[^MAB;1(qx;!)] = 0: (163) The matrices ^MAB for any A and B are all based on ^MTT, see appendix A.2.1. It is shown in appendix A.2.4 that the zeroes of the determinants of the other matrices are given by the zeroes of the determinant of ^MTT. Hence, only ^MTT is discussed here.

One knows that an arbitrary square matrix^Aof dimensionNN with elements a^ij is not singular when

ja^iij>^X

j⁶=i

ja^ijj; (164)

for any i with 1 i N [293, 294]. Now, the diagonal elements of the matrix

M

TT

;1 and also the sum over the o-diagonal elements of ^MTT;1 are estimated in appendix A.2.4, see eqs. (292) and (294). Due to the presence of poles in the corresponding expressions, the diagonal elements and the sums over the o-diagonal elements diverge when ! and qx are chosen such that they either lie exactly on the light or on the Luttinger dispersion;qy andqz may not be chosen freely but are xed by the discretization procedure. Away from these poles, the diagonal elements of

M

TT

;1 are roughly of the order of 1 while the sums over the o-diagonal elements are roughly of the oder of

Z = 4e^{2 2~}0c vF

c ^310;2 vF

c : ⁽¹⁶⁵⁾

For a realistic value of the Fermi velocity around 10⁵ m/s, Z 10^;5. Hence, only when ! andqx are chosen close to the light or to the Luttinger dispersion, the sums

58 4 CLEAN QUANTUM WIRE over the o-diagonal elements of ^MTT;1 become large enough in order to be equal to or larger than the diagonal elements { then, the matrix ^MTT;1 can be singular and thus local-eld eigenmodes can exist. For values of ! and qx away from the light and the Luttinger dispersion, local-eld eigenmodes cannot exist.

The combinations of ! and qx for which ^MTT;1 is singular determine the dis-persion relation of the eigenmode of the local eld. Now, the discussion in the preceding paragraph implies that, if eigenmodes exist at all, their dispersion rela-tion is closely related to the light and the Luttinger dispersions. This is in analogy to the dispersion relation of the phonon-polariton { the local-eld eigenmode in a system consisting of a ionic lattice and an electromagnetic eld {which was discussed in section 2.2.3. Thus, near the crossing of light and Luttinger dispersions, where the coupling between charge excitations and electromagnetic eld is strongest, the opening of a gap is expected.

In analogy to the phonon-polariton, the local eld eigenmode studied here is called plasmon-polariton as it is based on the coupling of a collective plasma oscillation of the charges in the quantum wire and an electromagnetic eld. The names \local-eld eigenmode" and \plasmon-polariton" are used synonymously in the following.

Note that the coupling of a plasma oscillation to a transverse eld is only possible due to the anisotropy of the system. In 3D isotropic systems, the plasma oscillation can only couple to a longitudinal eld [3].

Now, the above discussion on when the matrix^MTT;1 can be singular and when it cannot be singular gives an idea on what should be expected for the local-eld eigenmodes in the quantum wire. However, it does not proof that ^MTT;1 does become singular at all. To ascertain that local-eld eigenmodes exist in the quantum wire, to study their dispersion relation and to investigate the gap in this dispersion is the remaining task of this section.

Plasmon and Photon Dispersions

The branches of the dispersion relation of plasmon-polariton are expected to lie close to the dispersion relations of the light and of the collective excitations of the Luttinger liquid. The greatest deviation of the polariton dispersion from the light and the Luttinger dispersions is expected to appear near the crossing of these two dispersions.

The dispersion relation of the photon for a~q-vector parallel to the wire is denoted by !Ph^k and is !Ph^k =cqx because ^j~Q^j= 0. When the ~q-vector is not parallel to the wire, the dispersion relation of the photon is denoted by !Ph^? and is, as ^j~Q^{j 6}= 0,

!Ph^? =c(q_2x+Q²)¹⁼². Now, the dispersion !Ph^k intersects the Luttinger dispersion

!LLtwice as a function ofqx, see gure 3. One point of intersection is atqx =q1^k = 0.

A second point of intersection exists because for a Coulomb potential the Luttinger dispersion is innitely steep at qx = 0 and crosses over to a linear behaviour vFqx

for large qx, see section 3.2, while the light dispersion for ^j~Q^j = 0 rises as cqx with c nite and c > vF. Following eqs. (50) and (53), one can estimate the position of the second crossing between the photon dispersion for~qparallel to the wire and the Luttinger dispersion to be at q2^k

p8e^;8=Z, where Z is given in eq. (165).

The light dispersion for a~q-vector not parallel to the wire intersects the Luttinger

4.4 Local Field Eects 59 dispersion only if the component of ~q perpendicular to the wire is not too large, see gure 3. For a small enough ^j~Q^j, also the light dispersion !Ph^? intersects the Luttinger dispersion twice. The positions of the intersection are denoted by q1^?

and q2^?. These two positions cannot be estimated analytically, but one knows q1^k < q1^? < q2^? < q2^k, see gure 3. The photon and Luttinger dispersions are displayed as functions of qx and ^j~Q^j in gure 4.

qxd

!d=c

q1^kd q1^?d q2^?d q2^kd

Figure 3: The Luttinger dispersion

!LL (solid), the photon dispersion !Ph^k

(dashed), a photon dispersion !Ph^?that intersects !LL (dotted), and a photon dispersion !Ph^? that does not intersect

!LL (dash-dotted). The shaded area il-lustrates the continuum of photon dis-persions with any possible Q. The pic-ture is to be understood qualitatively.

0 qxd

0 Qd

Figure 4: The Luttinger dispersion (light grey) and the photon dispersion (dark grey) as functions of qxd and Qd. The picture is to be understood qualita-tively.

Polariton Dispersion Approached Analytically

One could now in principle determine the eigenmode dispersions numerically by searching for the zeroes of the determinant of ^MTT;1 using a numerical algorithm, see below. However, the region of the crossing between light and Luttinger disper-sions, which is the most interesting regime here as it promises the largest deviation of the eigenmode dispersion from light and Luttinger dispersions, cannot be reached numerically for realistic values of the Fermi velocity. But, in order to obtain a qual-itative picture, it can be approached analytically in the limit of very few abscissas.

Choose N = 3 and M = 1, evaluate ^MTT;1, take the determinant and look for the zeroes. One obtains three branches of the polariton dispersion, see appendix A.2.4,

!T;LL(qx)d c

2

!LL(qx)d c

2

1^; e^;3=2Z 72

; (166)

60 4 CLEAN QUANTUM WIRE

!T;Ph^k(qx)d c

2

(qxd)²+ 2Z

3 ; (167)

!T;Ph^?(qx)d c

2

(qxd)²+ (43;1)² + e³⁼²Z

6 ; (168)

whereZ is given in eq. (165). The value of ^j~Q^jd is here forN = 3 and M = 1 given by 43;1 = 2^p6. These three expressions in (166) - (168) are not valid for any value of qx. However, for the following discussion the conditions imposed on eqs. (166) -(168) are not important, details can be found in appendix A.2.4.

The dispersions (166) - (168) are displayed qualitatively in gure 5. The plasmon-like mode!T;LLlies below the Luttinger dispersion!LL. The dierence between!T;LL

and !LL is given by the multiplicative factor [1^;e^;3=2Z=72], it depends mainly on vF=c, see eq. (165), and increases with increasingvF=c. The photon-like modes!T;Ph^k

and !T;Ph^? lie above the photon dispersions!Ph^k and!Ph^?. The dierence between

!T;Ph^k and!Ph^k is given by the additive term 2Z=3 and the dierence between!T;Ph^?

and !Ph^? is given by the additive term e³⁼²Z=6 { these dierences depend on Z and hence onvF=c. Near the crossing of the dispersions!LL and!Ph^k, the branches !T;LL

and!T;Ph^kcross without showing any repulsion. This is due to the fact that for!Ph^k

the ~q-vector is parallel to the wire and hence the corresponding transverse eld is perpendicular to the wire. It has no component parallel to the wire that could couple to the plasmons. At qx = 0, however, the branches of the polariton dispersion !T;LL

and !T;Ph^k do show an anti-crossing because for ^j~q^j = 0 the transverse eld is not well dened and can have any direction. The repulsion between the branches!T;Ph^k

and !T;LL at qx = 0 is

!T;Ph^k(0)d c

2

;

!T;LL(0)d c

2

= 2Z

3 : (169)

The size of the gap at qx = 0 increases with increasing vF=c. This ratio determines the strength of the coupling between charges and electromagnetic elds.

qxd

!d=c

Figure 5: The Luttinger dispersions

!LL (solid), the photon dispersions !Ph^k

(long dashes) and !Ph^? (short dashes), and the corresponding branches of the eigenmode dispersion !T;LL (dash-dot),

!T;Ph^k (dots), and!T;Ph^?(dash-dot-dot) displayed qualitatively. The deviations of the eigenmode branches from the dis-persions of the uncoupled systems are greatly exaggerated.

In the eigenmode dispersion derived above for N = 3 and M = 1, no gap was found at nite qxd as !LL and !Ph^k do not couple at nite qxd due to symmetry reasons and !LL and !Ph^? do not cross, see gure 5. The reason for the absence

4.4 Local Field Eects 61 of a crossing between !LL and !Ph^? is that the value of ^j~Q^jd, which is xed by the discretization procedure, is too large. In the limit of innitely many abscissas, there are of course discrete values of Q small enough in order to allow for an intersection of !LL and !Ph^? { but that limit cannot be approached analytically. In order to be able to study the gap at a crossing of !LL and !Ph^? for nite qxd nevertheless, chooseN =M = 1. But then, the abscissasN=1;1 =M=1;1= 0, and hence^j~Q^j= 0, thus ~qis parallel to the wire. Now, shift N=1;1 slightly to the left and to the right, splitting it into two abscissas: N=1;1 ^! ", where 0 < " 1. The choice " ⁶= 0 ensures ^j~Q^{j 6}= 0 and hence the presence of !Ph^? with !Ph^?d=c = (q2xd² + 16"²)¹⁼². The explicit expressions for the branches of the ensuing eigenmode dispersion can be found in appendix A.2.4. They are denoted by!T;+ and!T;^;. It is impossible to call either of them photon-like or plasmon-like as such an assignment would inevitably depend of qxd. At qxd= 0, it is, for the conditions see appendix A.2.4,

!T(qx = 0)d c

2

(4")²+Z;

0: ⁽¹⁷⁰⁾

Here, in the regime where qxd is much smaller than the position of the rst crossing between!LLand!Ph^?,qxdq1^?d,!T^;is a plasmon-like mode and!T+is a photon-like mode. As "² Z, the relative deviation of the dispersion of the photon-like mode from the dispersion !Ph^? is huge at qxd = 0. The result in eq. (170) is not equivalent to the one in eq. (167) as in (170) the deviation is given by Z, while it is given by 2Z=3 in (167). This dierence is due to the crude approximations performed taking so few abscissas. However, in both equations the deviation is determined by Z and hence by vF=c!

At the two intersections of the dispersions !LL and !Ph^?, namely at qxd = qi^?d for i= 1;2, dene

id:= (qi^?d)²+ (4")² =

!LL(qi^?)d c

2

: (171)

Here, a gap opens and one obtains in lowest order in "²

!T(qx =qi^?)d c

2

( (id)²+Z ₍^(4")_id)²2;

(id)²: ⁽¹⁷²⁾

In the regimeq1^?d < qxd < q2^?d,!T^;is a photon-like mode while!T+ is a plasmon-like mode in contrast to the regime qxd q1^?d. For qxd q2^?, !T^; is again a plasmon-like mode and !T+ is again a photon-like mode, see appendix A.2.4. As

"=(id) < 1, the deviation of !T+ from the photon dispersion !Ph^? is smaller at qxd = qi^?d than at qxd = 0 and it is further smaller at q2^?d than at q1^?d. The deviation of !T^; from the Luttinger dispersion is of higher order in "² than the expression given in eq. (172) and thus much smaller than the deviation of the photon mode from !Ph^?. The size of the gap between the two branches of the eigenmode dispersion at qi^?d is in lowest order in "²

!T+(qx =qi^?)d c

2

;

!T^;(qx =qi^?)d c

2

=Z ⁽⁴")²

(id)²: (173)

62 4 CLEAN QUANTUM WIRE It is smaller at q2^?d than at q1^?d and increases with increasing vF=c. The results are displayed qualitatively in gure 6.

qxd

!d=c

Figure 6: The Luttinger dispersion

!LL (solid), the photon dispersion

!Ph;^? (dashes), and the corresponding branches of the eigenmode dispersion,

!T+ (dash-dot) and !T^; (dots), dis-played qualitatively. The deviations of the eigenmode dispersions from the dis-persions of the uncoupled systems are greatly exaggerated.

Note that a gap in between the branches of the polariton dispersion opens only if the corresponding dispersion relations of the light and the Luttinger liquid cross.

The crossing at qxd = 0 is independent of the shape of the interaction potential but a crossing at nite qxd depends crucially on the form of Vee. Apart from the fact that choosing a priori a nite-range interaction potential is not consistent with Maxwell equations, the light and the Luttinger dispersion do not cross at nite qxd if the interaction potential is not an innitely-ranged Coulomb potential. In fact, the light and the Luttinger dispersion cross only because the slope of the Luttinger dispersion forqxd^!0 is innitely steep which is due to the innite range of the Coulomb potential. For a nite-range potential, the Luttinger dispersion rises linearly for smallqxd, see section 3.2, with a slope smaller than the slope of the light dispersion and hence the two dispersions never cross. A gap in between the branches of the dispersion relation of the plasmon-polariton at nite qxd in analogy to the phonon-polariton is hence only present when a Coulomb potential is considered.

In contrast to the plasmon in a 3D homogeneous system discussed in section 3.3, here one does not nd a gap in the dispersion that exists for any value of the wave vector. This is due to the fact that the dispersion relation of the 1D plasma oscillation starts at zero frequency for a zero wave vector.

Polariton Dispersion Approached Numerically

From the above analytical approaches forN = 3,M = 1 andN =M = 1, it is clear what to expect qualitatively for dispersion relation of the plasmon polariton near the crossing of light and Luttinger dispersion. Nothing is yet known about the regime qxdq2^kd. In the following, this regime is probed numerically: The determinant of

M

TT

;1 is evaluated for arbitraryqx and!, thenqxand !are varied in order to nd a zero of det[^MTT;1]. The search for the zeroes of the determinant of ^MTT;1 is concentrated on the regime close to the light and the Luttinger dispersions as away from these dispersions ^MTT;1 cannot be singular, see above. As the determinant has to be evaluated quite often during the search for zeroes, this search becomes very slow for N;M >9.

4.4 Local Field Eects 63 It is found that the determinant of ^MTT;1 has zeroes for qx and ! close to the Luttinger dispersion relation, see gure 7. As the determinant exhibits sign changes in connection with these roots, this mode is quite easy to nd and can be evaluated to a relative accuracy as small as 10^;10. The mode is called plasmon-like as for qxdq2^kd it shows only plasmon like behaviour. It lies below the Luttinger dispersion, !T;LL(qx)< !LL(qx). The exact quantitative result for this plasmon-like mode depends on the number of abscissas chosen. The relative deviation of the Luttinger dispersion from the corresponding branch of the eigenmode is dened as

LL(qx) = !LL(qx)^;!T;LL(qx)

!LL(qx) ; (174)

see gure 8. The denition was chosen in order to ensure LL > 0. The relative deviation LL is of the order 10^;6. It increases with increasing N;M, but gure 8 indicates a convergence of LLwith increasingN;M. Further, the relative deviation LL decreases with increasing qxd. This is due to the fact that with increasing qxd the dispersion relations of the light and of the Luttinger liquid move further apart and hence the coupling between charges and electromagnetic eld becomes weaker.

0 0.002 0.004 0.006

0 1 2 3 4 5

qxd

!T;LL

Figure 7: The Luttinger dispersion !LL

(solid) and the numerically found eigen-mode !T;LL for N = M = 9 () and vF=c = 10^;3. The smallest value for qxd is 10^;4.

-7.8 -7.4 -7.0 -6.6 -6.2

0 1 2 3 4 5

qxd log10(LL)

Figure 8: The logarithm (base 10) of the relative deviation of the plasmon-like mode from the Luttinger dispersion for vF=c = 10^;3 and N = M = 3 (+), N =M = 5 (), N =M = 7 (+), and N =M = 9 ().

The light dispersion is here denoted by !Ph;nm(qx), it also depends on N and M, and follows from !Ph^? by replacing qy = 4N;n=d and qz = 4M;m=d. When N;n and M;m are both equal to zero, the light dispersion !Ph;nm corresponds to

!Ph^k, otherwise !Ph;nm corresponds to !Ph^?. Hence, one obtains a dispersion that correspond to !Ph^k when n = m = 2 for N = M = 3, when n = m = 3 for N = M = 5, when n =m = 4 for N =M = 7, when n =m = 5 for N = M = 9, and so on. For all other combinations of n;m and N;M, the dispersion !Ph;nm

corresponds to!Ph^?where the explicit value of^j~Q^jddepends on these combinations,

64 4 CLEAN QUANTUM WIRE

j~Q^jd =d[(4N;n)² + (4M;m)²]¹⁼². Further, the choice of the abscissas depends on a rule given in appendix A.2.1.

Now, forqxand!close to these light dispersions!Ph;nm, zeroes of the determinant of ^MTT;1 are found which are not always accompanied by a change of sign: A zero is sometimes related to an extremum which is zero at its smallest (for a minimum) or its largest (for a maximum) value. These zeroes are dicult to nd. Further, one sometimes obtains even two or more zeroes lying very close together. These double or triple zeroes are attributed to the nite number of abscissas and are assumed to merge into one single zero forN;M ^!1, see also the discussion in appendix A.2.4.

In the dispersion relations displayed in the following, one of the zeroes was selected in order to represent the eigenmode dispersion relation and consequently the exact quantitative behaviour of this dispersion depends on the choice of the zero. The overall qualitative behaviour, however, is independent of this choice. The relative accuracy lies roughly in between 10^;10 and 10^;6. The branch of the eigenmode whose dispersion lies in the vicinity of a light dispersion is called photon-like.

Despite the encountered diculties, the mode !T;Ph;nm(qx) is always present and lies slightly above the corresponding light dispersion, !T;Ph;nm(qx) > !Ph;nm(qx).

The relative deviation of the photon-like branch from the photon dispersion is Ph;nm(qx) = !T;Ph;nm(qx)^;!Ph;nm(qx)

!Ph;nm(qx) : (175)

The denition was chosen in order to ensure Ph;nm>0. Four photon dispersions for dierent discrete values of Q atN =M = 7 and the corresponding branches of the eigenmode dispersion are displayed in gure 9. The photon dispersion withn=m= 4 for N =M = 7 in gure 9 corresponds to !Ph^k. The relative deviations between photon dispersions and photon-like modes are shown in gure 10, but only for the photon modes which correspond to !Ph^?. The deviation decreases with increasing qx and with increasing Q, i.e. with increasing distance of the light dispersion from the Luttinger dispersion indicating that the coupling between charge excitations and electromagnetic elds decreases. Figures 11 and 12 display the relative deviation for the photon mode that corresponds to !Ph^k for N;M = 3, 5, 7, 9. The data in the two gures is the same, the only dierence is that qxd is displayed logarithmically in gure 12 but not in gure 11 and that the range of qxd is smaller in gure 12 than in gure 11. The deviation is much larger than the one displayed in gure 10 and even diverges for qxd ^! 0. This is due to the fact that Ph;nm in eq. (175) contains a !Ph;nm in the denominator which goes to zero with qxd^!0 for the data in gures 11 and 12. Further, the deviation decreases with increasing N;M but shows a tendency to convergence.

In gures 11 and 12, the relative deviation of the photon-like eigenmode from the photon dispersion diverges for qxd^!0 because !Ph;nm^!0. As an alternative, consider the absolute deviation, i.e. Ph;nm!Ph;nm, which should remain nite for qxd ^! 0. In gure 13, this absolute deviation of the photon-like mode !T;Ph;nm

that corresponds to !T;Ph^k from the photon dispersion that corresponds to !Ph^k is displayed. This absolute deviation indeed converges to a nite value for qxd ^! 0.

That the absolute deviation does not vanish with qxd ^! 0 indicates that !T;Phnm

4.4 Local Field Eects 65

0 2 4 6 8 10 12

0 1 2 3 4 5

qxd

!T;Ph;nm

Figure 9: The photon dispersions!Ph;nm

(lines) and the numerically obtained branches of the eigenmode !T;Ph;nm

(symbols) for N =M = 7 and n=m= 4 (dash-dotted, ), n = 4,m = 5 (solid, +), n = 4, m = 7 (long dashes, ), n = 5, m = 5 (short dashes, +) n = 5, m = 6 (dotted,) for vF=c = 10^;3 .

-7.4 -7.2 -7 -6.8

0 1 2 3 4 5

qxd log10(Ph;nm)

Figure 10: The logarithm (base 10) of the relative deviation of the photon-like mode from the photon dispersion for vF=c = 10^;3, N = M = 7, and n = 4, m = 5 (+), n = 4, m = 7 (), n = 5, m = 5 (+) n= 5, m= 6 ().

-6 -4 -2 0 2

0 1 2 3 4 5

qxd log10(Ph;nm)

Figure 11: The logarithm (base 10) of the relative deviation of the photon-like mode that corresponds to !T;Ph^k from the photon dispersion that corresponds to!Ph^k forvF=c = 10^;3 andN =M = 3 with n = m = 2 (+), N = M = 5 with n = m = 3 (), N = M = 7 with n = m = 4 (+), N = M = 9 with n =m= 5 ().

-4 -3 -2 -1 0 1 2

0.001 0.01 0.1

qxd log10(Ph;nm)

Figure 12: Same data as in gure 11 but on a logarithmic scale in qxd and in the regime 10^;4 < qxd <10^;1.

Im Dokument Frequency-dependent electronic transport in quantum wires (Seite 62-74)